Grothendieck’s trace map for arithmetic surfaces via residues and higher adèles

Morrow, Matthew

doi:10.2140/ant.2012.6.1503

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Grothendieck's trace map for arithmetic surfaces via residues and higher adèles

Matthew Morrow

Vol. 6 (2012), No. 7, 1503–1536

DOI: 10.2140/ant.2012.6.1503

Abstract

We establish the reciprocity law along a vertical curve for residues of differential forms on arithmetic surfaces, and describe Grothendieck’s trace map of the surface as a sum of residues. Points at infinity are then incorporated into the theory and the reciprocity law is extended to all curves on the surface. Applications to adelic duality for the arithmetic surface are discussed.

Keywords

reciprocity laws, higher adèles, arithmetic surfaces, Grothendieck duality, residues

Mathematical Subject Classification 2010

Primary: 14H25

Secondary: 14B15, 14F10

Milestones

Received: 26 June 2011

Revised: 25 October 2011

Accepted: 13 December 2011

Published: 4 December 2012

Authors

Matthew Morrow
Department of Mathematics University of Chicago 5734 University Avenue Chicago, IL 60637-1546 United States
http://math.uchicago.edu/~mmorrow/

Vol. 6, No. 7, 2012